Characterizing an L-Band Pulse Amplifier with Local FFT and Wavelet Transforms

Lorenzo
Carbonini has 12 years of experience in RF/MW
design and testing. He did achieve the Physics degree
cum laude at the University of Turin in 1989, then the
Mathematics degree cum laude at the same university in
1992. From 1989 to 1996 he has been RF/EMC design
engineer at Alenia Avionic Equipment Division. From
1996 to 1998 he has been Product Development Manager
for EMC testing products at Thermo Voltek Europe. In 1998 he joined Marconi Mobile covering different positions, he
is presently the Material Engineering Manager. He is the inventor of three
patents in the RF Design field.

The subject of this article is a technique
for characterizing the transient behavior of pulse power
amplifiers (PPAs). The technique is based on signal processing
applied to time-domain waveforms sampled directly at RF using a
fast digital storage oscilloscope (DSO). Spectral properties of
the output signal can be estimated locally on the waveforms,
and can indicate the correct strategy to fix the distortion.
The paper defines and discusses the local Fourier transform
(LFT) and wavelet transform (WT), and analyzes an L-band 150 W
PPA working at 1025 MHz. Signal processing information was used
to find and fix the distortion generation mechanism of a
spurious 700 MHz signal. LFT proves to be more effective than
WT and provides an effective insight into the spurious
generation mechanism.

Introduction

The RF designer developing power amplifiers often faces the
occurrence of spurious amplifier responses"the amplifier output
signal is not an amplified replica of the input signal but
exhibits substantial differences (distortion). These
differences may be associated with a number of causes,
overlapping in several cases. Among the possible causes of
distortion are: amplifier non-linearity, oscillations due to
the coupling of the RF transistor with input/output matching
networks, and oscillations in the power supply lines.

Signal distortion must be below some specified level. Since
this distortion often cannot be efficiently analyzed by
simulation, you must identify and eliminate it during the
experimental phase of the design.

Instruments typically available to the RF designer are a
power meter (PWM) and a spectrum analyzer (SA). The spectrum
analyzer is particularly efficient in determining the
occurrence of distortions since, in several cases, these
distortions lead to spectral deformation of the output signal
or to spurious spectral lines.

The output spectrum of a power amplifier is subjected to
various constraints, including those associated with the
maximum level of harmonics or of undesired spectral lines, or a
spectral mask limiting the spectrum degradation close to the
carrier frequency fc.

When dealing with pulse power amplifiers (PPAs), the
situation becomes more complex due to the abrupt variations in
the input and output signal. For these cases, the information
provided by PWMs and SAs is not sufficient to investigate, in
depth, the phenomena causing distortion. The peak power meter
(PPWM) is a useful instrument in this case, measuring the
instantaneous signal power at a (repetitive) sampling rate of
several Megahertz. You can use the PPWM to study the signal
amplitude distortion with a dynamic range of about 30 dB.

Recently, equipment vendors have introduced real-time SAs
with the capability of performing instantaneous spectrum
measurements over a bandwidth of several MHz (for example, the
Tektronix 3000 series). This kind of instrument exhibits a wide
dynamic range, but is not sufficient to analyze, in detail,
narrow pulses (for example, shorter than 10 ms).

Digital sampling oscilloscopes (DSOs) with a 8 to 16 GS/s
non-repetitive sampling rate and input analog bandwidth of 1.5
to 2 GHz have been introduced. Examples of these instruments
include the Agilent Infiniium 54845A and Lecroy WavePro 960.
This kind of instrument has the advantage of capturing the
signal amplitude and phase variations directly at RF, even for
very narrow pulses. The main limitation of this solution is the
achievable dynamic range (normally about 40-50 dB) which is
determined by the DSO's ADC. The measurement technique proposed
in this article is based on a numerical analysis of waveforms
sampled by a DSO"a technique particularly well suited to
investigate the physical mechanisms underlying signal
distortion. As a consequence, an adequate analysis of the
sampled waveform can be valuable to the RF designer in
determining the reason for a distortion and in implementing
adequate countermeasures.

Throughout this article, an example using an L-band PPA
illustrates the DSO waveform-sampling technique.

Distortion in Power Amplifiers

Consider the case of an amplitude modulated (AM) signal
sin(t) at the input of a power amplifier, where:

(1)

ain(t) is the time-dependent signal amplitude,
wc = 2pfc is the angular frequency,
and fc is the carrier frequency.

In the general case, the signal at the power amplifier
output is:

(2)

In Equation 2, both aout(t) and f(t) depend on the present and all the
preceding values of ain(t) (dependence on previous
values indicates a "memory effect" of the system).

A physical constraint on the form of sout(t) is
that aout(t) = 0 if t<tg,
tg>0, which is the group delay. In other
words, the amplifier output is zero before the input signal is
applied and propagated through the amplifier. Another condition
is that f(t) =
wc
tg + t w
1(t), where w1(t)
is an instantaneous frequency variation.

The mechanisms causing distortion may be very complex and a
general treatment is very difficult. However, two main
phenomena can occur:

Linear Distortion
In this case Sout(w) =
F(w) Sin(w), where F(w) is a transfer function. This
distortion occurs mainly across the signal frequency band,
but also spectral lines out of band may occur if the transfer
function exhibits poles at values of
w very close to the real axis. The effects of linear
distortion are shown in Figure 1, and result in the
occurrence of resonance frequencies (wspur) and time-domain
waveform distortion.

Non-Linear Distortion
In the particular case of a memory-less system with
polynomial transfer function of degree M, the output signal
is .
This implies that the output spectrum is:

(3)

The coefficient a1 is
equivalent to the small signal gain;
implies the occurrence of the second harmonic;
implies the occurrence of the third harmonic together with
third order inter-modulation distortion; and so on. The
effect of non-linear distortion is summarized in Figure
2. This distortion results mainly in spectrum enlargement
close to the carrier and occurrence of harmonics in the
frequency domain, together with signal clipping with sharper
rise and fall times in the time domain.

Figure 1: Input and output signal for linear
distortion

Figure 2: Input and output signal for non-linear
distortion

In practice, what frequently happens is that the amplifier
outputs spurious frequencies due to linear distortion and
inter-modulation products between the input spectrum. You can
model these phenomena as the effect of a linear distortion
passing through a memory-less non-linear distortion. The effect
is shown in Figure 3, where the time-domain output signal
exhibits linear distortion and signal clipping.

However, the simplified models this article has outlined
have limited validity. Another effect is phase distortion due
to the amplifier's nonlinear behavior. This distortion results
in amplitude modulation (AM) to phase modulation (PM)
conversion, and a spurious phase modulation is visible at the
amplifier output. This effect is particularly visible on the
rising and falling edges of the waveform.

Another effect is power-supply-line oscillation during the
rise and fall of the waveform. This effect is particularly
prevalent in Class C amplifiers since the amplifier current
increases abruptly when the RF input signal is present.

Linear amplitude distortion may result in some in-band
distortion and is the cause of spurious spectral lines.
Linear phase distortion results in, essentially, in-band
effects.

Non-linear phase distortion causes a variation of the
instantaneous carrier frequency, hence contributing to peak
broadening. This type of distortion occurs evenly during rise
and fall times of the input waveform.

Power-supply-line oscillation occurs during the waveform's
rise and fall times and may be different during the rise and
fall times.

Figure 3: Input and output signal for complex
distortion

Measurement Technique and Test Results

As explained in the introduction, there are DSOs available
with sampling rates of several GS/s. This kind of instrument is
well suited for sampling signals in the RF/MW range, so that
both amplitude and phase can be directly measured on the RF
waveform. This analysis is not possible with PPWMs and SAs.

The main drawback of DSOs is the low sensitivity normally
available, due to the range of the instruments' input ADCs.

The main parameters to be considered when evaluating DSOs
for measuring RF signals are:

The oscilloscope sampling rate fs

Number of bits Nb of the ADC.

In order to correctly sample the input signal one should
have, according to the sampling theorem, the relation fs>
2fMAX must hold, assuming that the signal spectrum
is confined below fMAX.

If a characterization of some harmonics of the input signal
is needed, the required sampling rate grows rapidly.

If the input signal contains high-frequency components which
cannot be properly sampled by the DSO, you should connect a
low-pass filter (LPF) with cut-off at fs/2 to the
oscilloscope input to avoid aliasing (under-sampling high-frequency components that may generate in-band noise).

The ADC number of bits Nb affects the DSO
instantaneous dynamic range. As a rule of thumb, the
theoretical dynamic range of a DSO is given as:

Normally the dynamic range is further limited by other
factors. A tradeoff is possible between the system's dynamic
range and the signal bandwidth.

Indeed, if the sampling rate is 22K times
fMAX, then by numerical filtering the number of bits
of the signal can be increased by a factor K (which means that
the dynamic range is increased by K 6.02 dB). This function
is often implemented in the DSO firmware.

The analyzed pulse was essentially a rectangular RF pulse
with rise time of about 200 ns, fall time of about 300 ns,
carrier frequency fc of 1025 MHz, and width of about
8 ms. It was the output signal of a RF amplifier with 150 W
peak output power; this amplifier was the driver stage of a 1
kW pulse amplifier.

Figure 4 shows a schematic of a typical bipolar class
C common base stage of a power amplifier. Min and
Mout are, respectively, the input and output matching
networks, Lin and Lout are inductances
(normally suitable microstrip lines) necessary to guarantee the
correct transistor polarization.

Figure 4: Schematic of a bipolar Class C amplifier
stage

The output signal was sampled at 4 GS/s in order to capture
the whole pulse width with a reasonably small output file. With
this sampling rate, a maximum signal frequency of 2 GHz would
be correctly sampled, exceeding the DSO input passband;
therefore, in this case no LPF was required.

The test results, which will be analyzed throughout this
article, are shown in Figures 5 and 6. These are output
signals sampled at the driver output. The first signal was
measured at the beginning of the amplifier optimization, and
includes a spurious resonance at 700 MHz; in the second signal,
this resonance was suppressed. The difference between the two
signals is negligible.

Transforms: Principles and Practical
Applications
Time-domain data measured by the DSO were sequences of 8-bit
samples with spacing Ts = 1/fs = 0.25 ns.
Transforms applied to the sampled signal highlight features
that are necessary to understand the physical phenomena.

The most important and well-known transform is the Fourier
transform (FT), in its computationally efficient form, the fast
Fourier transform (FFT). FTs are important because they present
a spectrum of the signal that would be obtained by an ideal
SA.

To compute the DFT of the signal according to Equation
4, you would need Ns2
multiplications, in principle. If Ns is a power of
2, the FFT algorithm can be applied and only Ns
log2Ns multiplications are necessary.

Two important aspects need to be pointed out. The first is
that Equation 4 should be applied to periodic signals,
with period Ns Ts (in other words, s(t)
= s(t + Ns Ts)). If this condition is
not met, you get in-signal aliasing, which means that the
spectrum of the sampled signal is not a true sampling of the
spectrum of the original signal. In the case of an isolated
pulse, this condition is easily met provided that the pulse is
well centered in the time window; in other word, there is some
integer ks for which
when k<ks or
k>Ns-ks. Furthermore, when
dealing with measured data some aliasing is present in any case
because of the quantization noise of the signal close to the
zero value.

The residual aliasing may be avoided by a proper windowing,
by defining a signal
where
if
ks<k<Ns-ks,
wk=0 if k<kt or
k>Ns-kt, with
kts.

The choice of the window wk must be such that it
minimally influences the signal spectrum. A good review of
classical windows used for FT is reported in.

The FFT technique has been applied to the measured signals
shown in Figures 5 and 6. The signals were truncated to
216 samples without any window, due to limited pulse
length, and then a FFT was performed. Figure 7 shows the
spectrum of the measured PA driver output signal before
optimization; a strong spurious resonance is present at 700 MHz
with level -25 dBc. Referring to the Distortion in Power
Amplifiers discussion, due to the nonlinear behavior of the
class C amplifier an inter-modulation product is visible at
1350 MHz with level -35 dBc. The main problem with this
distortion was that the final stages of the power amplifier
were particularly sensitive to the resonance, which even
resulted in a reduction of output power.

Note that the spectra obtained by a simple FFT are similar
to what can be obtained by SA tests: as such, the information
is global and is not related to the details of the time-domain
waveform. Hence no indication about the distortion mechanism is
provided by the measurement and the only possible analysis is
purely experimental.

Localized Fourier Transform
Detailed knowledge of the signal spectral properties along
different portions of the waveform, such as rise or fall edges,
can be valuable in understanding the physical mechanism
of distortion in pulse power amplifiers.

This can be accomplished quite easily by applying a
localized FT (LFT) to the signal. You can define the LFT as
follows:

Define a time internal of Nw samples over
which the localized spectrum is computed

Choose a proper window wk (k = 0, ..,
Nw-1) to smooth the data

Choose a time step of Nst Ts for
the spectral analysis.

Defining tl = l ts Nst,
the result of a localized LFT is the following:

(5)

SLFT may be defined as a spectrogram
(time-dependent spectrum).

Nw determines the detail to which the spectral
analysis is performed, in other words, the interval over which
the FFT is performed. A large value of Nw implies a
low resolution in the time domain and high resolution in the
frequency domain, and vice versa. Nst determines the
granularity in the time domain of the spectral estimate.

The figure was obtained based on a 216 points
signal, Nw = 29, and Nst =
394. The spectrum granularity is 7.8125 MHz, the time window is
128 ns wide, and the window is a squared raised cosine with
expression:

(6)

It is evident from Figure 9 that the resonance at 700 MHz
is not a transient close to the transistor rise and fall edges.
Due to this conclusion, resonant features of the supply lines
can be excluded along with S-parameter phase variations due to
the varying input power.

The attention was then focused on the input and output
matching networks of the transistor. Figure 10 shows that a
resistor placed in parallel with the input inductance Lin was
sufficient to suppress the resonance. The inter-modulation at
1350 MHz, due to the amplifier non-linearity, was also
reduced.

Figure 10: Spectrogram of the driver output signal
at 1025 MHz after the spurious elimination

The spectrogram analysis shows also that the major part of
other spurious responses occur solely at the pulse's falling
edge, a memory effect. These responses are probably due to the
transient response of the power supply lines when the
amplifier's current goes from a peak level to zero.

The spectrograms show also a slight increase of the spurious
levels along the pulse around about 750-800 MHz. This effect is
probably due to a temperature rise in the transistor die and is
a further memory effect in the system.

Wavelet Analysis
As shown in the preceding section, the use of transforms with
local features can be quite useful for the analysis of
transient signals. A relatively recent technique is represented
by the wavelet transform. The following discussion only covers
basic wavelet theory. Interested readers can consult the
references for a deeper introduction to the subject (for a
practical introduction and a wider list of references, see ).

The wavelet transform (WT) technique and synthesis is well
suited for a variety of problems in which standard Fourier
analysis is inadequate or impractical. Applications include
image processing, signal compression, noise reduction, and
digital transmission techniques.

Wavelet theory is a functional analysis technique in which a
signal is projected on a basis of functions with compact or
almost compact support (in other words, functions that are
essentially localized). In the case of Fourier analysis, the
basis function upon which the signal is projected is not
localized (ejwt); hence,
wavelet techniques are particularly well suited to analyze the
transient features of signals. Moreover, signals with abrupt
variations and/or discontinuities in the derivatives exhibit a
very broad Fourier spectrum. You can construct wavelets, which
exhibit discontinuities in the derivatives, and describe in a
more efficient way these kinds of signals.

Wavelet analysis is based on a wavelet function denoted as
y(t), satisfying some regularity conditions. Different wavelets have
been found in the literature. The wavelet we will use in the
analysis is the complex Morlet wavelet
yM(t):

(7)

Looking at the wavelet shapes, it is evident from
Equation 7 that the Morlet wavelet is very similar to a
windowed Fourier basis function.

Given a wavelet y(t), the
corresponding wavelet transform of the signal s(t) is:

(8)

The parameter a, which must be strictly positive, is the
scale of the signal, while the parameter b is the signal's
position. a is somehow related with the frequency of the signal
(although only Fourier transforms provide rigorous
frequency-content information) and defines, at the same time,
the width of the region over which the signal analysis is
performed. b identifies the position at which the analysis is
performed. A discrete version of Equation 8 may also be
formulated. The frequency can be associated to the parameter a
by the following formula:

(9)

In Equation 9, the frequency fy is defined as the frequency of the
wavelet, in other words the frequency at which the wavelet FT
reaches a maximum. For the Morlet wavelet, .

We are now ready to interpret the results of wavelet
transforms of the signals of Figures 11 and 12.

As shown in Figures 11 and 12, the differences in the
Morlet wavelet transforms before and after spurious elimination
are negligible; in particular, the resonance at 700 MHz (scale
of about 2.86) is not very evident.

Conclusions

This article has illustrated a technique for characterizing
the transient behavior of PPAs. The technique is based on
numerical processing applied to time-domain waveforms sampled
at RF using a fast DSO.

The advantage of this technique is that spectral properties
can be estimated locally on the waveforms, providing considerable insight into the physical mechanism at the basis
of signal distortions. This is a considerable advantage over
standard SA measurements and reduces the need for long
experimental optimization time (normally performed by trial and
error).

Some of the possible causes of signal distortion, including
linear and nonlinear distortion and supply line oscillation
have been described in this article.

The article also defined and discussed two different
transform techniques, LFT and WT; in the WT case, we used the
Morlet and order 2 Daubechies wavelets.

An L-band 150 W PPA working at 1025 MHz was analyzed. The
distortion generation mechanism of a spurious signal at 700 MHz
was found, and the information available through numerical
signal processing suggested a strategy for suppressing the
resonance (working on the input/output matching networks).

LFT proves to be more effective than WT in analyzing the RF
signals and provides an effective insight into the spurious
generation mechanism.